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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > 1arymaptfv | Structured version Visualization version GIF version |
Description: The value of the mapping of unary (endo)functions. (Contributed by AV, 18-May-2024.) |
Ref | Expression |
---|---|
1arymaptfv.h | ⊢ 𝐻 = (ℎ ∈ (1-aryF 𝑋) ↦ (𝑥 ∈ 𝑋 ↦ (ℎ‘{〈0, 𝑥〉}))) |
Ref | Expression |
---|---|
1arymaptfv | ⊢ (𝐹 ∈ (1-aryF 𝑋) → (𝐻‘𝐹) = (𝑥 ∈ 𝑋 ↦ (𝐹‘{〈0, 𝑥〉}))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq1 6881 | . . 3 ⊢ (ℎ = 𝐹 → (ℎ‘{〈0, 𝑥〉}) = (𝐹‘{〈0, 𝑥〉})) | |
2 | 1 | mpteq2dv 5241 | . 2 ⊢ (ℎ = 𝐹 → (𝑥 ∈ 𝑋 ↦ (ℎ‘{〈0, 𝑥〉})) = (𝑥 ∈ 𝑋 ↦ (𝐹‘{〈0, 𝑥〉}))) |
3 | 1arymaptfv.h | . 2 ⊢ 𝐻 = (ℎ ∈ (1-aryF 𝑋) ↦ (𝑥 ∈ 𝑋 ↦ (ℎ‘{〈0, 𝑥〉}))) | |
4 | eqid 2724 | . . . . 5 ⊢ (0..^1) = (0..^1) | |
5 | 4 | naryrcl 47530 | . . . 4 ⊢ (ℎ ∈ (1-aryF 𝑋) → (1 ∈ ℕ0 ∧ 𝑋 ∈ V)) |
6 | 5 | simprd 495 | . . 3 ⊢ (ℎ ∈ (1-aryF 𝑋) → 𝑋 ∈ V) |
7 | 6 | mptexd 7218 | . 2 ⊢ (ℎ ∈ (1-aryF 𝑋) → (𝑥 ∈ 𝑋 ↦ (ℎ‘{〈0, 𝑥〉})) ∈ V) |
8 | 2, 3, 7 | fvmpt3 6993 | 1 ⊢ (𝐹 ∈ (1-aryF 𝑋) → (𝐻‘𝐹) = (𝑥 ∈ 𝑋 ↦ (𝐹‘{〈0, 𝑥〉}))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 Vcvv 3466 {csn 4621 〈cop 4627 ↦ cmpt 5222 ‘cfv 6534 (class class class)co 7402 0cc0 11107 1c1 11108 ℕ0cn0 12470 ..^cfzo 13625 -aryF cnaryf 47525 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5276 ax-sep 5290 ax-nul 5297 ax-pr 5418 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-id 5565 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-ov 7405 df-oprab 7406 df-mpo 7407 df-naryf 47526 |
This theorem is referenced by: 1arymaptf1 47541 |
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